Sets, Relations & Functions
Set Operations
Grade 11
Question:
<p>Given <span class='formula'>X = \{ n \in \mathbb{N} : 1 \leq n \leq 50 \}</span>, <span class='formula'>A = \{ n \in X : n \text{ is multiple of } 2\} = \{2, 4, 6, 8, \ldots, 50\}</span>, and <span class='formula'>B = \{ n \in X : n \text{ is multiple of } 7\} = \{7, 14, 21, 28, 35, 42, 49\}</span>. Find the smallest subset of <span class='formula'>X</span> containing elements of both <span class='formula'>A</span> and <span class='formula'>B</span>.</p>
Step-by-Step Solution
Key Concept: Use the inclusion-exclusion principle to find the cardinality of the union of two sets: |A ∪ B| = |A| + |B| - |A ∩ B|.
<p><strong>Solution:</strong></p><p>The smallest subset of <span class='formula'>X</span> containing elements of both <span class='formula'>A</span> and <span class='formula'>B</span> is <span class='formula'>A \cup B</span>.</p><p>We note that <span class='formula'>14, 28, 42 \in A \cap B</span> (elements divisible by both 2 and 7).</p><p>Using the principle of inclusion-exclusion:</p><p><span class='formula'>n(A \cup B) = n(A) + n(B) - n(A \cap B) = 25 + 7 - 3 = 29</span></p><p>∴ The answer is <strong>29</strong>.</p>
Correct Answer: 29